Related papers: Links and dynamics

We prove that any link in $S^3$ whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either $\mathbb{Z}$ or…

This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invariants of classical knots and links. More…

We derive the $s$-invariants of certain simply connected $7$-manifolds whose second homology groups are isomorphic to $\mathbb{Z}^{2}$. We apply the $s$-invariants to give a partial classification of simply connected total spaces of circle…

In this chapter (Chapter V) we present several results which demonstrate a close connection and useful exchange of ideas between graph theory and knot theory. These disciplines were shown to be related from the time of Tait (if not Listing)…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points…

Machine learning methods are being actively considered as a new tool of describing many body physics. However, so far, their capabilities has been only demonstrated in previously studied models, such as e.g. Ising model. Here, we consider a…

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we…

We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves…

Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the…

Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams…

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the $\phi$-mapping topological current theory. The main purpose in this paper is to present a new theoretical framework which can directly give the…

Minimal flow spaces of dimension 1 are among the most fundamental limit sets in dynamical systems. These invariant sets occur as the typical minimal sets in surface flows, the minimal sets of suspensions of subshifts (for example, in Lorenz…

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first…

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting…

The statistical properties of random lattice knots, the topology of which is determined by the algebraic topological Jones-Kauffman invariants was studied by analytical and numerical methods. The Kauffman polynomial invariant of a random…

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister…

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology…

Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems,…

In this paper we present a detailed study of \emph{bonded knots} and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or…